Download Scattering model for quantum random walks on a hypercube

PHYSICAL REVIEW A 71, 012306 (2005)
Scattering model for quantum random walks on a hypercube
Jozef Košík1 and Vladimír Bužek1,2
1
Research Center for Quantum Information, Institute of Physics, Slovak Academy of Sciences, Dúbravská cesta 9,
845 11 Bratislava, Slovakia
2
Faculty of Informatics, Masaryk University, Botanická 68a, 602 00 Brno, Czech Republic
(Received 22 March 2004; published 4 January 2005)
Following a recent work by Hillery et al. [Phys. Rev. A 68, 032314 (2003)], we introduce a scattering model
of a quantum random walk (SQRW) on a hybercube. We show that this type of quantum random walk can be
reduced to the quantum random walk on the line and we derive the corresponding hitting amplitudes. We
investigate the scattering properties of the hypercube, connected to the semi-infinite tails. We prove that the
SQRW is a generalized version of the coined quantum random walk. We show how to implement the SQRW
efficiently using a quantum circuit with standard gates. We discuss one possible version of a quantum search
algorithm using the SQRW. Finally, we analyze symmetries that underlie the SQRW and may simplify its
solution considerably.
DOI: 10.1103/PhysRevA.71.012306
PACS number(s): 03.67.Lx, 05.40.Fb
I. INTRODUCTION
Quantum random walk is a theoretical concept conceived
to simulate certain algorithms using quantum-mechanical elements, i.e., unitary operators and measurements [1]. In particular, it has been shown recently that it is possible to use a
quantum random walk to perform a search in a database with
the topology of the hypercube faster than it can be done
classically [2]. It is an oracle-based algorithm, which is optimal in its speed. Another successful application of quantum
random walks has been demonstrated by Childs et al. [3],
who have also constructed an oracle problem that can be
solved by a quantum algorithm exploiting a quantum random
walk exponentially faster than any classical algorithm.
These two examples justify the general hope that quantum
random walks might be able to solve some problems, based
on random processes [4] (e.g., Monte Carlo methods, 2-SAT,
graph connectivity, etc.) faster than corresponding classical
algorithms.
In general, there are essentially three types of quantum
random walks. First, let us mention the so-called coined
quantum random walk (CQRW), which is a discrete time
walk which makes use of an additional quantum system, the
coin [5]. The second type of quantum random walks is described by a continuous (Hamiltonian) dynamics of a quantum system [6]. Quantum random walks on regular graphs
have been first discussed by Watrous [7]. The third type of
quantum random walks based on physical model of optical
multiports has recently been introduced by Hillery et al.
[8,9].
Before we proceed, we note that a quantum random walk
as discussed by Aharonov et al. [10] is basically a mapping
␺ : GV → Cd, which is updated at each step by a function
␺共x兲 哫 F关␺共y兲 : 共xy兲 苸 GE兴, where G = 共GV , GE兲 is a graph with
vertices GV and edges GE.
We therefore can say that the quantum random walk is a
special instance of the quantum cellular automaton [11,12].
The classical cellular automaton is a concept general enough
to accommodate virtually any algorithm; more precisely, any
1050-2947/2005/71(1)/012306(8)/$23.00
Turing machine can be simulated using a cellular automaton.
The CQRW is usually defined on regular graphs (each
vertex having the same number of outgoing edges). The definition on nonregular graphs is also possible, and some interesting algorithms are based on this version [13]. However,
the latter version does not possess the symmetries of the
former one, nor its neat tensor product structure (the unitary
evolution operator CQRW on the regular Cayley graph commutes with generators of the underlying group). Instead, the
whole graph must be addressed, by means of an oracle which
tells us whether any two vertices are connected by an edge
[14], which causes a considerable growth of the resources.
In this paper, we will focus our attention on a quantumoptical model of multiports [8,9] which describes a possible
physical realization of specific quantum random walks. In
this scheme, we have an array of multiports (see, e.g., [15]
and references therein), interconnected with optical paths. A
photon is launched into one path and is transformed by the
action of the multiports. This action can be described as a
scattering process, therefore we will refer to this scheme as
the scattering quantum random walk (SQRW).
The SQRW is more viable from the experimental point of
view, can be extended to nonregular graphs, and is equivalent to CQRW on the regular graphs.
We will investigate a particular arrangement of the multiports when are localized at the vertices of a hypercube. The
array of multiports effectively acts as a scattering potential,
when connected to semi-infinite tails. It can be endowed with
two characteristic values, namely the reflection and transmission amplitude for photons. This was done in Refs. [8,9] for
a special two-dimensional hypercube.
Our paper is organized as follows. In Sec. II we define the
SQRW on the hypercube. In Sec. III, we show how the
SQRW on the hypercube may be reduced to the cellular automaton on the line. In addition, we will compute the hitting
amplitude and we will make some other simulations. In Sec.
IV, we will investigate scattering properties of the hypercube
connected to semi-infinite tails. Section VII contains the
proof that the SQRW is equivalent to a generalized version
of the coined quantum random walk. In Sec. VI, we show
012306-1
©2005 The American Physical Society
PHYSICAL REVIEW A 71, 012306 (2005)
J. KOŠÍK AND V. BUŽEK
FIG. 1. Action of the multiport 共U兲 on the ingoing photon (in)
on the dim 3 hypercube. Coherent superposition of three photonic
excitations is created.
how to implement the SQRW efficiently on the quantum circuit. In Sec. VIII, we discuss one possible version of a search
algorithm using the SQRW. Finally, in Sec. IX, we will analyze symmetries that underlie the SQRW and which may
considerably simplify the solution of the model.
where Ux is isometry Hx → Ĥx (onto, hence linear, hence
unitary). Since the subspaces are orthogonal, U is unitary.
The concrete realization of the unitary operator Ux reflects
the fact that the multiport partially reflects and partially
transmits the ingoing photon (see Fig. 1). Denoting the reflection and transmission coefficients r and t, respectively,
we have
Ux兩yx典 = r兩xy典 + t
兺
共xz兲苸GE
兩xz典,
共2.1兲
where 共xy兲 苸 GE. For U to be unitary, these coefficients must
satisfy the relations [8]
共2.2兲
共d − 2兲兩t兩2 + r * t + rt * = 0.
共2.3兲
The operator Ux (for any x) has in the natural basis the matrix of the form 共Ux兲ij = r␦ij + t共1 − ␦ij兲. The relations (2.2)
may be satisfied by using the Grover coefficients r = 共2 / d兲
− 1, t = 2 / d (in what follows, we will call this setup the
Grover multiport; for more details, see Ref. [16]).
The pseudo-eigensystem of any such Ux can be readily
computed (“pseudo” means that we set an isomorphism between Hx and Ĥx such that 兩yx典 ⬅ 兩xy典). Since Ux = 2兩s典具s兩
− 1 (兩s典 is the complete superposition over the basis of the
domain of Ux, which we denote 兩1典 , . . . , 兩d典), the eigenvectors
are 兩s典 (with pseudo-eigenvalue 1) and linearly independent
(but not orthogonal) set 兵兩1典 − 兩k典 : k = 2 , . . . , d其 [with
共d − 1兲-degenerate pseudo-eigenvalue −1]. Performing the direct sum of these eigenvectors leads us to a pseudoeigensystem of U, with pseudo-eigenvalues ±1.
There are many other choices of the reflection and the
transmission coefficients. One set of them is the following
(we will use it when necessary):
t=
II. THE DEFINITION OF SQRW
The SQRW was first presented in Ref. [8]. The technique
behind its implementation is the multiports [15]: linear optical elements, interconnected with optical paths. Each multiport has in general a different number of inputs (which at the
same time serve also as outputs). A coherent superposition of
photons entering the multiport is transformed into another
coherent superposition of photons outgoing from the multiport. The multiports are the vertices of a graph GV, and the
optical paths are its edges.
The photon traveling between the multiports x , y is denoted 兩xy典. The Hilbert space on which the state of the photon is defined is spanned by vectors 兩xy典, 共xy兲 苸 GE and can
be decomposed into a direct sum of Hilbert subspaces H =
丣 x苸G Hx, where Hx = span兵兩yx典 : 共yx兲 苸 GE其. That is, the basis
V
of Hx is composed of the states of photons ingoing into the
multiport at x. For convenience, we introduce the Hilbert
subspace spanned by the states of photons outgoing from the
vertex at x: Ĥx = span兵兩xy典 : 共xy兲 苸 GE其.
The evolution operator of states in H is U = 丣 x苸GV Ux,
兩r兩2 + 共d − 1兲兩t兩2 = 1,
r=
冑
1
,
dp
1−
d − 1 i␪
e ,
d2p
共2.4兲
where cos ␪ = 共1 − d / 2兲 / 冑d2p − d + 1 with p ⬎ 1 / 2. If we set
p = 1, then the reflection amplitude goes to limd→⬁r = − 21
+ i共冑3 / 2兲. Obviously, for large dimensions, almost all of the
photons will be reflected. The multiports with these coefficients will be the so-called symmetric multiports. We will
later compare both sets of coefficients with respect to their
mixing properties.
From now on, we will be dealing with the multiports arranged into the form of the d-dimensional hypercube. The
edges of the hypercube will form two-way optical paths.
The d-dimensional hypercube is the Cayley graph G
= 共Zd2 , 关d兴兲, where 关d兴 = 兵0 , . . . , d其 is the set of generators of
the additive (mod 2) group Zd2 (the binary strings with only
one 1). For any x , y 苸 Zd2, we set the scalar product xy
= x1y 1 + . . . + xdy d (mod 2). The norm 兩x兩 = 冑xx is the Hamming weight (the number of 1s in x). The set ᐉw = 兵x : 兩x兩
= w其 is called the layer of the hypercube.
For simplicity, we will denote the basis states of H for the
d-hypercube 兩xy典 as 兩x ; a典, where x is the vertex and a
= 1 , . . . , d is the generator such that x + a = y.
III. THE TOPOLOGY OF THE HYPERCUBE
The hypercube may be “broken up” in individual layers
ᐉw, i.e., sets of vertices with equal Hamming weight. There is
a special class of vectors from H, which is closed with respect to U and whose members may be described by a
smaller number of coefficients, thus simplifying the evolution equations. Namely, these are the vectors 兩␺典 such that
012306-2
PHYSICAL REVIEW A 71, 012306 (2005)
SCATTERING MODEL FOR QUANTUM RANDOM WALKS…
具x ; a 兩 ␺典 is the same for all 兩x兩 = w and for all a. Under this
assumption, the vectors are specified by coefficients 兵␺w,±其
where ␺w,± = 具x ; a 兩 ␺典 with 兩x兩 = w, 兩x + a兩 = w ± 1.
The reduced equations for evolution of the coefficients
␺w,± are given from the assumptions that each vertex from ᐉw
has a fixed number of edges which connect it to the previous
and next layer (with Hamming weight having w ± 1). We note
that a vertex 兩x兩 = w is connected with w edges from the previous layer and with d − w edges with the next layer. Since
the coefficients assigned to the projection 兩␺典 to a given layer
are fixed, we obtain the recursive relations,
共U␺兲w,+ = tw␺w−1,+ + 关t共d − w − 1兲 + r兴␺w+1,− ,
共U␺兲w,− = t共d − w兲␺w+1,− + 关t共w − 1兲 + r兴␺w−1,+ . 共3.1兲
For w = 0 and w = d, these equations still hold wherever it
makes sense, i.e., for ␺0,+ , ␺d,−. The coefficients ␺−1,· , ␺d+1,·
are neglected as long as they are multiplied by zero in the
equation.
The evolution which is governed by these equations is
called the symmetric SQRW.
If the initial state is ␺0,+ = 1 / 冑d, then we immediately obtain the expression for the hitting amplitude
␺d,−共d兲 = 关t共d − 1兲 + r兴共d − 1兲!
td−1
冑d .
共3.2兲
Using the Grover parameters (r = 2 / d − 1 and t = 2 / d), we find
for the quantum probability pq to get from the vertex 兩x兩 = 0 to
the vertex 兩x兩 = d in d steps the expression
pq = 兩␺d,−共d兲兩2 =
冉冊
d!
dd
2 d−1
4
d
.
共3.3兲
Classically, the probability that we are at a given vertex
from ᐉw is pw, for which holds 共Wp兲w = 共1 / d兲wpw−1 + 共1 / d兲
⫻共d − w兲pw+1 − 1. From this we obtain the hitting probability
共Wd p0兲 = 共1 / dd兲d!. The classical probability to get from the
vertex 兩x兩 = 0 to vertex 兩x兩 = d in d steps reads
pc =
d!
.
dd
4d−1
.
d
unitary. But we need the unitarity only to conserve the inner
product 具␣ជ 兩 ␣ជ 典. Actually, the inner product
ជ 兩␺ជ 典 =
具␺
兺冉 冊
d
w=0
d
w
2
共兩␺w,+兩2 + 兩␺w,−兩2兲
共3.7兲
is conserved.
Though we have simplified the problem by the assumption of symmetric initial values, we are still far from its
explicit solution. The solution would rely on the path integration along different paths by which two sites can be connected in a presupposed number of steps. Each path would
be assigned a complex amplitude (basically some product of
r , t), and by adding all the relevant paths together, we would
get the amplitude distribution over the hypercube. This normally gives us enormous combinatorial expressions, which
are difficult to interpret.
Since the SQRW on the hypercube with symmetric initial
states is equivalent to the nonunitary one-dimensional (quantum) random walk on the finite sequence of layers ᐉw, we
may explore the probability distribution pn共w兲 over the layers for an initial state 兩␺0典 = 兺a共1 / 冑d兲兩0 ¯ 0 ; a典, where
pn共w兲 is the expectation value of the operator
Mw ª
共3.4兲
Using the previous expression, we find that the classical and
quantum hitting probabilities are related like
pq = p2c
FIG. 2. The ratio pq / pc of hitting probability for classical 共pc兲
and quantum 共pq兲 random walk on the hypercube, related to the
dimension d.
兺
兩x;a典具x;a兩,
共3.8兲
x苸ᐉw,a
i.e., pn共w兲 = 具␺0兩共U†兲nM wUn兩␺0典. In Fig. 3, we present a result
of simulation of the evolution of pn共w兲.
共3.5兲
The ratio pq / pc as a function of the dimension d of the hypercube is given by the expression
pq d! 4d−1
,
=
pc dd d
共3.6兲
and is plotted in Fig. 2. The whole model can be used to
simulate the coined quantum random walk on the line segment with the position-dependent coin. The state of the coin
is described by a vector 关␺w,+ , ␺w,−兴, w = 0 , . . . , d and is updated at each step with the update rules given by Eq. (3.1).
We might be troubled by the fact that the update rule is not
FIG. 3. The probability distribution of the SQRW, pn共w兲, for the
hypercube of the dimension d = 50, with a symmetric initial state
localized at the vertex 0 ¯ 0, and steps n = 1 , ¯ , 100. We consider
Grover multiports.
012306-3
PHYSICAL REVIEW A 71, 012306 (2005)
J. KOŠÍK AND V. BUŽEK
FIG. 4. The probability distribution of the symmetric SQRW on
the hypercube of dim 50, for steps n 0 to 250, with the initial state
␺0,+ = ␺d,− = 1 / 冑2d (other ␺ ’ s = 0) and symmetric multiports.
FIG. 6. The probability distribution of the symmetric SQRW on
the hypercube of dim 50, for steps n 0 to 250, with the initial state
␺0,+ = ␺d,− = 1 / 冑2d (other ␺ ’ s = 0) and Grover multiports.
We see that the SQRW on the hypercube is effectively
isomorphic to a dynamics of a resonator: We start with one
excited site, and the excitation propagates as a Gaussian
packet along the resonator. As the packet hits the boundary,
the corresponding site is excited, and the packet is reflected
in the opposite direction.
From Fig. 3, we see that the hypercube acts as a resonator
when it comes to the evolution of the probability distribution
over the layers.
We have performed detailed simulations for the symmetric SQRW with initial conditions such that either only the
coefficients ␺0,+ ; ␺d,− are nonzero, or ␺d/2,± ⫽ 0. There are
two choices of the multiports: the Grover and the symmetric
multiports. We make the simulations for both of them, and
for both initial conditions (see Figs. 4–7). We see the periodicity of the evolution, another feature akin to the resonant
behavior. Note that the Grover multiport has much better
mixing properties than the symmetric multiport, owing to the
fact that it is more distant from unity.
weights 0 and 2 has been studied. Each tail has been supposed to be a one-dimesional lattice with perfectly transmitting multiports. Along one tail, a photon enters the hypercube, and emerges on the other side. It is possible to
calculate explicitly the transmission coefficient of the whole
structure. In the present section, we will study an analogous
problem for an arbitrary-dimensional hypercube. We will utilize some symmetry assumptions, which allows us to perform the calculation (or at least the simulation) for arbitrary
dimensions. The scheme we consider is shown in Fig. 8.
Now everything is as before, except that the multiports at
the vertices with Hamming weight 0 and d have reflection
and transmission coefficients given by expressions r̃ = 关2 / 共d
+ 1兲兴 − 1 and t̃ = 2 / 共d + 1兲, respectively. The multiports outside
the hypercube are perfectly transmitting. The initial state is a
photon traveling from the vertex −1 to the vertex 兩x兩 = 0 of
the hypercube. The state of the whole system is described by
complex numbers ␣w,± which represent the amplitude that
the photon travels from a vertex 兩x兩 = w and is directed to the
next or previous layer ᐉw±1. Now we let w run through Z.
The resulting relations are
IV. HYPERCUBE AS A SCATTERING POTENTIAL
In Ref. [8], a model of a two-dimensional hypercube with
semi-infinite tails attached to the vertices with the Hamming
FIG. 5. The probability distribution of the symmetric SQRW on
the hypercube of dim 50, for steps n 0 to 250, with the initial state
d
兲 (other ␺ ’ s = 0) and symmetric multiports.
␺d/2+1,± = 1 / 冑2共 d/2+1
共U␺兲0,+ = t̃␺−1,+ + 关共d − 1兲t̃ + r̃兴␺1,− ,
FIG. 7. The probability distribution of the symmetric SQRW on
the hypercube of dim 50, for steps n 0 to 250, with the initial state
d
兲 (other ␺ ’ s = 0) and Grover multiports.
␺d/2+1,± = 1 / 冑2共 d/2+1
012306-4
PHYSICAL REVIEW A 71, 012306 (2005)
SCATTERING MODEL FOR QUANTUM RANDOM WALKS…
d
兩 ␺ 0典 =
FIG. 8. The scattering potential (three-dimensional hypercube).
The vertices outside the hypercube are denoted −⬁ , . . . , −2 and d
+ 1 , . . . for the hypercube of dimension d.
␥ j兩0 ¯ 0; j典.
兺
j=1
共5.1兲
Now the amplitude of Ud兩␺0典 to project onto the state
兩1 ¯ 1;⫹典 is given by the sum of amplitudes to traverse from
·0 to 1 ¯ 1 in d steps, which is t共d−1兲t̃. In particular, for the
initial state 兩0 · ; j典 we have 共d − 1兲! such paths. The situation
is analogous for all j, with each initial direction j contributing the factor ␥ j of the amplitude. The overall amplitude is
d
具1 ¯ 1; + 兩Ud兩␺0典 =
共U␺兲0,− = r̃␺−1,+ + dt̃␺1,− ,
␥ j共d − 1兲!t共d−1兲t̃.
兺
j=1
共5.2兲
The probability of detecting the particle at the state
兩1 ¯ 1;⫹典 depends only on 兺dj=1␥ j. In this sense, the hypercube with tails attached behaves like a Mach-Zehnder interferometer.
共U␺兲d,+ = dt̃␺d−1,+ ,
共U␺兲d,− = 关共d − 1兲t̃ + r̃兴␺d−1,+ ,
VI. IMPLEMENTING THE SQRW
共U␺兲w,+ = tw␺w−1,+ + 关r + 共d − w − 1兲t兴␺w+1,− ,
共U␺兲w,− = t共d − w兲␺w+1,− + 关t共w − 1兲 + r兴␺w−1,+ . 共4.1兲
We begin with the particle in the state 兩−1 , 0典, i.e., a particle localized at the vertex just left of the hypercube on the
tail, and pointing to the right (see Fig. 8).
We have simulated the probability that a particle incoming from the left will be absorbed by the detector after n
steps (see Fig. 9). This means that the system evolves for n
steps, and then a projection on the vector 兩d ; d + 1典 is performed. The result shows periodic beats of the probability of
the absorption of the photon by the detector.
V. NONSYMMETRIC INITIAL STATE
When we impose a symmetry condition on initial states,
the whole problem becomes linear in the dimension of the
hypercube (normally its complexity is exponential in d). But
we also may be interested in the behavior that appears as a
result of phase differences between the components of the
initial state. Given the d-dimensional hypercube with semiinfinite tails attached to the vertices 0 ¯ 0 and 1 ¯ 1, we
consider the initial state
FIG. 9. The scattering probability of a 10-dimensional hypercube, for a photon incoming from the source (S) (n is the number of
steps). The multiports are symmetric.
Until now, we have not discussed the question whether it
is feasible to implement the SQRW. To build a whole network of multiports, we need exponentially growing resources (the number of vertices grows exponentially). However, to encode the states under consideration, we need only
ddlog de qubits. So we can ask a question: Is it possible to
build a network of quantum gates operating on the qubit
register of this size? This is most easily done only on the
hypercube without the semi-infinite tails attached; however,
it is also possible to implement this scheme by adding some
overhead of gates to the network. We need d qubits for the
position register 兩x典 and at least dlog de qubits for the direction register 兩␸典. The first part of one application of the unitary operator U is controlled negation of each bit of x depending on 兩␸典. The second part is the transformation of the
state 兩␸典, so that the action of the multiports is accounted.
More precisely, the first part is
兩x典兩␸典 →
and the second part reads
兺a 兩x + a典具a兩␸典兩a典,
冋
兩x典兩a典 → 兩x典 r兩a典 +
兺 t兩b典
b⫽a
册
,
共6.1兲
共6.2兲
for each a = 1 , . . . , d. The first part described by Eq. (6.1) can
be implemented using a variant of the controlled-NOT (CNOT)
gate. The CNOT gate operates on two qubits such that it negates the first (target) qubit, iff the second (control) qubit is
nonzero. The action of the CNOT gate is described by the
two-qubit operator CCNOT = ␴x 丢 兩1典具1兩 + 1 丢 兩0典具0兩. We employ the ␾CNOT gate, which differs from the CNOT gate in that
it has a d-dimensional control state, unlike a single qubit. If
the control state is 兩␾典 (the accepting control state), then the
target qubit is negated, otherwise it is kept in the original
state. The operational form of the ␾CNOT gate is
␾CNOT = ␴x 丢 兩␾典具␾兩 + 1 丢 共1 − 兩␾典具␾兩兲.
Obviously, the ␾CNOT is unitary.
012306-5
共6.3兲
PHYSICAL REVIEW A 71, 012306 (2005)
J. KOŠÍK AND V. BUŽEK
it is translationally invariant. We search for the eigenvectors
in the form ␺k = 兺a=1,de2␲ika/dea, which yields
共6.7兲
where ␭k = r − t if k ⫽ 0, and ␭k = r − t + td.
FIG. 10. The gate which implements the SQRW on the
d-dimensional hypercube. The input state is the position register (d
qubits labeled as 1 , . . . , d) and the direction register 兩␸典. There are d
␾CNOT gates stacked together, with accepting states 兩1典 , . . . , 兩d典,
which change the position register, and the gate M which changes
the direction register.
The operation (6.1) can be implemented by using d ␾CNOT
gates (see Fig. 10). Each gate operates on a different qubit
from the position register. If the gate operates on the ath
qubit, the accepting control state is chosen to be 兩a典 (see Fig.
10).
The operations (6.1) and (6.2) are implemented by using a
single unitary operation M operating on the direction register. It corresponds to the transformation of the state due to
the multiports. It reads
M=
兺a
冋
r兩a典具a兩 + t
兺 兩b典具b兩
b⫽a
The d ⫻ d matrix form of M is
M=
冢
r
t
t
⯗
r
t ¯
¯ t
¯
t
r
冣
册
共6.4兲
.
共6.5兲
.
Consequently, the unitary evolution operator U of the SQRW
on the hypercube may be decomposed as U = G2G1 = 共1
丢 M兲C1 ¯ Cd, where Ca = ␴x 丢 兩a典具a兩 + 1 丢 共1 − 兩a典具a兩兲 is the
␾CNOT operator with the target qubit being the ath qubit from
the position register, and accepting the state 兩␾典 = 兩a典, with
a = 1 , . . . , d. The operators Ca are mutually commuting, and
have common eigenvectors. To find them, we decompose the
eigenvectors into the product of 共2 ⫻ d兲-dimensional vectors
兩␺典 丢 兩␹典. Applying Ca on 兩␺典兩␹典, we obtain (providing that
␴x兩␺典 = ␭兩␺典)
VII. SQRW IS SUPERSET OF COINED QUANTUM
RANDOM WALK
In this section, we will discuss the connection between the
scattering and the coined quantum random walks. The
SQRW reduces to the coined quantum random walk on a
regular graph (having all vertices with the same degree), and
conversely, the SQRW is the generalization of the coined
quantum random walk on general graphs.
There is an isomorphism between the coined quantum
random walk (CQRW) and the SQRW on the same Cayley
graph over the Abelian group, G. We recall that the CQRW is
defined by a unitary operator E on the Hilbert space HE
= HX 丢 HA, where HX is spanned by vectors 兩x典 , x 苸 GV, and
HA is spanned by the generators of G, the basis vectors 兩a典.
One step of the CQRW is given by E = SC, where S = 兺aTa
丢 ␲a and C = 1 丢 M. Here Ta兩x典 = 兩x + a典 is the translation, ␲a
is the projection to 兩a典, and M is any unitary operator. The
isometry is given by one-to-one mapping of basis vectors of
both H and HE like 兩x共x + a兲典H ⬅ 兩x典兩a典HE. The correspondence between operators U and E is Ux兩yx典, where y + a = x is
the same as applying the translation S on 兩x − a典兩a典 and then
applying the coin M on 兩a典 such that matrix representations
of Ux in the natural basis of Hx , H̃x and M in the natural
basis of HA are the same.
For regular graphs, we can decompose HE into the direct
product of HEx such that HEx = span兵兩x − a典兩a典 : a is gen.G其.
The scheme for generalizing the coined quantum random
walk on general graphs was proposed in Ref. [14], but this
has required an oracle which operates on the set of all edges
of the graph. Our scheme is based on local operations done
by multiports, so it is more reasonable and easier to implement physically. This was actually proposed in Ref. [17].
Algorithms based on coined quantum random walks were
proposed in Ref. [13].
Ca兩␺典兩␹典 = ␴x兩␺典兩a典具a兩␹典 + 兩␺典 丢 共兩␹典 − 兩a典具a兩␹典兲
= 兩␺典 丢 关共␭ − 1兲具a兩␹典兩a典 + 兩␹典兴
=
再
兩␺典兩␹典,
␭ = 1,
兩␺典 丢 关兩␹典 − 2具a兩␹典兩a典兴, ␭ = − 1.
冎
VIII. SEARCHING WITH SQRW
共6.6兲
The case with ␭ = −1 has to be dealt with separately. If 兩␹典
= 兩a典, then we get Ca兩␺典兩a典 = −兩␺典兩a典, and if 兩␹典 = 兩b典 , b ⫽ a,
then Ca兩␺典兩b典 = 兩␺典兩b典. We arrive at the conclusion that the
eigensystem of Ca is the set of vectors 兩␺典兩␹典, where 兩␺典 is
the eigenvector of ␴x and ␹ = 兩a典, a = 1 , . . . , d. What about the
eigensystem of G2? The matrix M can be diagonalized, since
In this section, we will address a question whether it is
possible to use the SQRW for a database search or a similar
task. To answer this question, we need to formulate what a
quantum database is and how we can move around its entries
using the SQRW.
The database we are searching in is the so-called quantum
dictionary. The classical dictionary is a set of pairs (key,
value). The set of all keys is given by the topology of a
graph, yielding the adjacency relations among all the keys.
012306-6
PHYSICAL REVIEW A 71, 012306 (2005)
SCATTERING MODEL FOR QUANTUM RANDOM WALKS…
Random walk (classical) in the dictionary is bound to the
edges of this graph.
The searching problem in the dictionary is given as follows: given a value, find a key, such that (key,value) is in the
dictionary. For N keys, this is an O共N兲 problem. To obtain a
quantum version of this scheme, we have to “quantize” (noncanonically) the problem. Due to the fact that the graph is
not regular, we cannot factorize the complete Hilbert space,
but we need to label the states in the most general fashion:
兩xy典, where 共xy兲 is an edge. The searching procedure consists
of applying one step of the SQRW, and then by querying the
database. The query corresponds to an application of a unitary operator (the oracle) [18], which flips some auxiliary
qubit, depending on whether the value assigned to the key is
the one we are searching for. That is, the oracle is the transformation O兩xy典兩q典 哫 兩xy典兩q 丣 f共x , y兲典, where f共x , y兲 gives the
value 1 if any of the vertices x , y satisfy the query, and 0
otherwise. It is clear that the oracle O is unitary. Now the
searching algorithm is based on the sequence of operations
共OU兲n, where U makes one step of the SQRW and O is the
oracle query (equivalent to the action of the multiports). One
such algorithm has been presented in Ref. [2]. In our terms it
is the SQRW on the hypercube, where the multiport assigned
to one marked key has trivial coefficients r , t (they only
change the phase), while the other multiports have coefficients corresponding to the action of the Grover operator to
the direction states. In Ref. [2], it has been shown that the
marked key can be found in O共冑N兲 steps with probability
O共1兲, where N is the number of vertices of the hypercube.
IX. SYMMETRIES OF THE EVOLUTION
OPERATOR U
The basic relation U␺ = ␭␺, where ␺ = 兺xa␥xa兩xa典 yields
the following recurrence relation:
r␥x,−a + t
兺 ␥xb = ␭␥x+a,a .
共9.1兲
b⫽a
Finding the symmetries of this operator helps us to find its
eigensystem. We can Fourier transform the states of H to
another basis, in which solutions can be found more easily.
The operator U has many symmetries, one of which is the
translation Tb : x 哫 x + b. The eigenvectors of Tb are (for details, see Ref. [16])
˜ 典=
兩ka
兺x 共− 1兲kx兩x;a典,
共9.2兲
˜ 典 is
with eigenvalues 共−1兲kb, k 苸 Zd2. The action of U on 兩ka
冉
˜ 典 = 共− 1兲ka r兩ka
˜ 典+
U兩ka
兺 兩kb˜典
b⫽a
冊
,
Ṽk =
¯
t共− 1兲kd
¯
t共− 1兲k1 r共− 1兲k2 t共− 1兲k3
⯗
t共− 1兲k1
¯
¯
r共− 1兲kd
冣
.
共9.4兲
Now we only need to find the eigensystem of this comparatively small matrix. It is obvious that Ṽk is translationally
symmetric. The eigensystem of Eq. (9.4) can be found in
Ref. [16].
In what follows, we will find another symmetry. Unlike
the previous case, now we will be changing both the elements of the position and the direction Hilbert spaces. This
transformation R will change the vector 兩x , a典 such that the
binary string x is cyclically shifted right by one place and a
is set to a 丣 1 modulo d. Since a is unambiguously defined
by the position in the binary string at which x differs from
x + a, this transformation is a symmetry. This can be viewed
as a rotation along the line segment connecting two opposite
vertices 0 ¯ 0 and 1 ¯ 1. We can choose any other two
vertices x , y such that 兩x − y兩 = d, and get a symmetry operator
Rxy = B†x RBx, where Bx changes the role of 0 to x and 1 to 1
+ x. More precisely, Bx兩z , a典 = 兩z + x , a典 (hence B† = B). Two
transformations Rx , Ry generally do not commute, but they
both commute with U.
X. CONCLUSION
We have proved that the SQRW is in fact a version of the
coined quantum random walk. We can use this observation to
extend the coined quantum random walk to the cases of nonregular graphs. While it is in principle easy to construct the
SQRW on any graph, it is still a question whether we also
can simulate it efficiently (e.g., like in Sec. VI). This point is
crucial for further development of quantum algorithms based
on the SQRW in higher dimensions (where the speedup may
become noticeable). The class of algorithms based on the
SQRW is the database searching, using the oracle queries
along with the “random” steps. We already know at least one
such algorithm (see Ref. [2]) and we know that it is optimal.
We cannot expect that the complexity drops below O共冑N兲 for
N database keys, but the new algorithms may be more general in their inputs, and may be easier to implement.
We have found the connection between mixing properties
of the multiport (or the coin) and the distance of the respective operator from unity. It might be interesting to find an
exact function of this distance, which yields the measure of
mixing for the SQRW.
共9.3兲
˜ 典, U has the form Ũ = diag共兵Ṽ 其兲, where
and in the basis 兩ka
k
冢
r共− 1兲k1 t共− 1兲k2
ACKNOWLEDGMENT
This work was supported by the European Union projects
QUPRODIS, QGATES, and CONQUEST.
012306-7
PHYSICAL REVIEW A 71, 012306 (2005)
J. KOŠÍK AND V. BUŽEK
[1] J. Kempe, Contemp. Phys. 44, 307 (2003).
[2] N. Shenvi, J. Kempe, and K. B. Whaley, Phys. Rev. A 67,
052307 (2003).
[3] A. Childs, R. Cleve, E. Deotto, E. Farhi, S. Gutmann, and D.
A. Spielman, in Proceedings of the 35th ACM Symposium on
Theory of Computing (STOC’03) (ACM Press, New York,
2003), pp. 59–68.
[4] We should note that there is nothing really random (in a classical sense) about quantum random walk: the quantum evolution is always unitary (deterministic), and the only randomness
involved in the scheme is the random outcomes of the measurements.
[5] Y. Aharonov, L. Davidovich, and N. Zagury, Phys. Rev. A 48,
1687 (1993).
[6] A. M. Childs, E. Farhi, and S. Gutmann, Quantum Inf.
Process. 1, 35 (2002).
[7] J. Watrous, J. Comput. Syst. Sci. 62, 376 (2001).
[8] M. Hillery, J. Bergou, and E. Feldman, Phys. Rev. A 68,
032314 (2003).
[9] E. Feldman and M. Hillery, e-print quant-ph/0403066.
[10] D. Aharonov, A. Ambainis, J. Kempe, and U. Vazirani, in
Proceedings of the 33rd Annual ACM Symposium on Theory of
Computing (STOC’01) (ACM Press, New York, 2001), pp. 50–
59.
[11] D. A. Meyer, J. Stat. Phys. 85, 551 (1996).
[12] B. Schumacher and R. F. Werner, e-print quant-ph/0405174.
[13] A. M. Childs and J. M. Eisenberg, e-print quant-ph/0311038;
see also A. Ambainis, e-print quant-ph/0311001.
[14] V. Kendon, e-print quant-ph/0306140; V. Kendon and B. Sanders, e-print quant-ph/0404043.
[15] I. Jex, S. Stenholm, and A. Zeilinger, Opt. Commun. 117, 95
(1995).
[16] C. Moore and A. Russel, e-print quant-ph/0104137.
[17] A. Ambainis, e-print quant-ph/0403120.
[18] M. A. Nielsen and I. Chuang, Quantum Computation and
Quantum Information (Cambridge University Press, Cambridge, 2000).
012306-8